Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations

• Autorzy: Osmolovskii N. P. , Maurer H.
• Języki publikacji: EN

Abstrakty

EN

In Part 1 of this paper (Osmolovskii and Maurer, 2005), we have summarized the main results on the equivalence of two quadratic forms from which second order necessary and sufficient conditions can be derived for optimal bang-bang control problems. Here, in Part 2, we give detailed proofs and elaborate explicit relations between Lagrange multipliers and elements of the critical cones in both approaches. The main analysis concerns the derivation of formulas for the first and second order derivatives of trajectories with respect to variations of switching times, initial and final time and initial point. This leads to explicit representations of the second order derivatives of the Lagrangian for the induced optimization problem. Based on a suitable transformation, we obtain the elements of the Hessian of the Lagrangian in a form which involves only first order variations of the nominal trajectory. Finally, a careful regrouping of all terms allows us to find the desired equivalence of the two quadratic forms.

Słowa kluczowe

EN

bang-bang control; second order necessary and sufficient conditions; critical cones; equivalence of quadratic forms; representation of the Hessian of the Lagrangian

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2007
• Tom: Vol. 36, no 1
• Strony: 5--45
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Osmolovskii N. P.

autor

Maurer H.

• Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland,

Bibliografia

• AGRACHEV, A.A., STEFANI, G. and ZEZZA, P.L. (2002) Strong optimality for a bang-bang trajectory. SIAM J. Control and Optimization 41, 991-1014.
• MAURER, H., BÜSKENS, C., KIM, J.-H.R. and KAYA, C.Y. (2005) Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optimal Control Applications and Methods 26, 129-156.
• MAURER, H. and OSMOLOVSKII, N.P. (2004) Second order sufficient conditions for time optimal bang-bang control problems. SIAM J. Control and Optimization 42, 2239-2263.
• MILYUTIN, A.A. and OSMOLOVSKII, N.P. (1998) Calculus of Variations and Optimal Control. Translations of Mathematical Monographs 180, American Mathematical Society, Providence.
• OSMOLOVSKII, N.P. (1988) High-order necessary and sufficient conditions for Pontryagin and bounded-strong minima in the optimal control problems. Dokl. Akad. Nauk SSSR, Ser. Cybernetics and Control Theory 303, 1052-1056, English transl., Sov. Phys. Dokl. 33 (12) (1988), 883-885.
• OSMOLOVSKII, N.P. (2004) Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Journal of Mathematical Sciences 123 (3), 3987-4122.
• OSMOLOVSKII, N.P. and MAURER, H.. (2005) Equivalence of second order optimality conditions for bang-bang control problems, Part 1: Main results. Control and Cybernetics 34, 927-950.
• PONTRYAGIN, L.S., BOLTYANSKII, V.G., GAMKRELIDZE, R.V. and MISHCHENKO, E.F. (1961) The Mathematical Theory of Optimal Processes. Fizmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.